Let . A class is investigating whether constitutes the general solution to the…

MATHEMATICS · Single correct · ISC · Mathematics · Calculus · Order and degree

Problem

Let F={y(x)=C1+C2x+C3ex∣C1,C2,C3∈R}\mathcal{F}=\{y(x)=C_1+C_2x+C_3e^{x}\mid C_1,C_2,C_3\in\mathbb{R}\}. A class is investigating whether F\mathcal{F} constitutes the general solution to the third-order differential equation y′′′−y′′=0y'''-y''=0. Which of the following provides the best justification for the conclusion? (a) Because the characteristic equation has a repeated root, one of the constants is lost; therefore only two arbitrary constants should appear, so F\mathcal{F} is too large to be the general solution. (b) Substituting yy into the differential equation yields −C1−C2x=0-C_1-C_2x=0, forcing C1=C2=0C_1=C_2=0; hence F\mathcal{F} cannot represent the general solution. (c) The family F\mathcal{F} contains three linearly independent arbitrary constants—matching the third order of the differential equation—and every member satisfies y′′′−y′′=0y'''-y''=0, so it is the general solution. (d) Because exe^{x} already includes the series terms 11 and xx, the constants on 11 and xx are redundant; at least four arbitrary constants are required for a fully general solution.

Related theory

Browse the problem library